The steady circulation-preserving hydrodynamic motions with velocity \({\mathbf q}=q{\mathbf t}\) are studied under either of the geometric constraints \({\mathbf t}\cdot\text{curl}{\mathbf t}=0\) or \(\text{div}{\mathbf t}=0\). It is established that the geodesic unit tangent \({\mathbf t}\)-field is constrained by privileged Heisenberg spin-type equations. In the case \(\text{div}{\mathbf t}=0\), remarkably, the integrable Heisenberg spin equation which is known to be equivalent to the solitonic nonlinear Schrödinger equation is obtained. It is remarked that the analysis presented in the paper is valid for the magneto-hydrostatics system wherein the magnetic field lines are geodesics on the surfaces containing the current density vector and magnetic field lines.